$\begingroup$How about the weak topology on a separable Banach space? It's not metrizable nor even first countable, so can't be homeomorphic to any subspace of $\mathbb{R}^\omega$.$\endgroup$
– Nate EldredgeNov 5 '18 at 15:17

$\begingroup$Well, there is probably a mistake in $\pi$-base, since the second space I gave is the box topology in $\mathbb{R}^{\omega}$, which is not connected (Steen-Seebach, Counterexamples in Topology, p. 128-129).$\endgroup$
– Francesco PolizziNov 5 '18 at 15:54

2

$\begingroup$If you actually click on the "connected" listing for that space, it says "this is wrong, please delete me". It seems that the Github code for $\pi$-base correctly says "false": github.com/pi-base/data/blob/master/spaces/S000107/properties/…. I don't know why they are out of sync. The "this is wrong" text doesn't even appear in the git history of that file.$\endgroup$
– Nate EldredgeNov 5 '18 at 22:51

Or take the long line: $\omega_1\times[0,1)$ with the lexicographic order, minus the first point. Every bounded open interval is isomorphic to $(0,1)$, so it is homogeneous. It is first-countable but not second-countable, hence not embeddable into $\mathbb{R}^\omega$.